We have motherly and fatherly. There is the city of brotherly love, and the girlfriend whose love for you is sisterly, or worse yet daughterly. According to the Collins Dictionary’s trend line, sonly was in use during the 18th century, but has flatlined since, and auntly enjoyed a brief period of use in the last half of the 20th century. I suppose Uncle Sam’s love for you is avuncular, but not unclely.
I’d say the most obvious difference between words like “mother,” “father,” “sister,” “brother,” and “daughter” vs. “son” and “aunt” is the stress pattern. The former group are all two-syllable words with the stress on the first syllable, so maybe there’s something about that stress pattern that better accommodates an “-ly” suffix.
That hypothesis has some counterexamples, though, such as the two-syllable words “uncle” (but not *uncle-ly), “parent” (but not *parent-ly), and “sibling” (but not *sibling-ly).
So maybe it’s a combination of the stress pattern and the word-final “-r”. (But then you have “cousinly,” which shows up in the OED a couple of times in the 19th century.) I still think that the /nl/ consonant cluster in “cousinly” is much less awkward than the potential /ntl/ cluster in “parent-ly” or /ŋl/ in “sibling-ly,” not to mention the double /l/ in “uncle-ly.”
So my final hypothesis, barring other counterexamples, would be a combination of stress patterns and consonant clusters.
There might also be a historical angle, though. I.e., does “parent” accept an “-ly” suffix less readily because it’s a borrowing from French/Latin? (Words like “mother” and “daughter,” as well as the “-ly” suffix itself, are of Germanic origin and have Germanic cognates such as “mütterlich.”)